Thursday, March 1, 2018

Axioms and Truth

Don’t worry: there will be no math.

Recently I’ve seen a number of posts and articles in which the writers try to talk about axioms and get it wrong. Therefore, this rant.

The writers will keep mis-defining axioms. To boil the definitions down to the simplest possible statement: “An axiom is a statement which is self-evidently true.”

Uh, no.

Axioms are more like rules of the game. For example, let’s look at some poker rules, because nobody confuses the rules for any type of poker to be self-evident truths, right? And poker is an easy example for me, because I learned it sitting under the kitchen table and sneaking beers while the nominal adults in the family bet and bluffed.
(Caveat: this is not intended as a complete set of instructions for any given type of poker; I’m trying to keep it down to the minimum necessary to prove my point.)

Five-Card Draw

Probably the simplest form of poker. Some of the rules are:

-Each player gets five cards
-Players may look at their cards
-There is a round of betting
-After the first betting round, each player may discard one to three cards face down and gets an equal number of cards, also face down, from the dealer.
-After all players have had a chance to draw, there is a second round of betting.

These are (some of) the axioms of Five-Card Draw. Note that none of them are self-evidently true; they’re just the rules of the game, and they can be changed to make variations on the game.

Deuces Wild

For instance, suppose you add a new axiom to those above:

-The four deuces (twos) are wild cards, which can be used as any card the holder needs to complete a hand (with one exception, which we don’t need to go into here).

This axiom isn’t “true” either, right? It’s just a new rule which makes for a slightly different game.

Everything’s Wild

You can always add to the number of wild cards by changing that first axiom of Deuces Wild. My relatives, after a sufficient number of beers have been consumed, have been known to play Deuces,Fives, and Jacks Wild, which makes, as you might say, a wild game.

But suppose you change that first rule to “All cards are wild cards.”

Presto, the game collapses. Now you are free to declare that all your cards are aces and show a hand of Five of a Kind, Aces, which would be a winning hand - except that everybody ese has the exact same hand.

Not surprisingly, this is an axiom which is never used.

Keeping it interesting

Mathematicians (okay, I lied just a tiny bit), just like poker players, like to work with sets of axioms that define an interesting set of possibilities. Sometimes these axioms appear to be obvious truths, like the rules of Euclidean geometry, which seem to be true statements about the world you can see. But pull back a bit, look at the whole world. It’s a sphere. And suddenly Euclid’s axioms don’t quite work. Your obvious truths… aren’t true any more.

And that’s why axioms are rules of the game, not self-evident truths.


  1. Ooo, a pet peeve, I do believe. ;-) But I don't blame you. There are certain things like drive me crazy as well, like the omission of "ly" on the end of some words - like saying safe instead of safely - a better example eludes me at the moment. (He made it around the track safe vs He made it around the track safely). I'm always adding it in my head, although when my husband was alive, we would both loudly add "LEE" when that mistake was made in a tv show. Anyway, I'm a firm believer that using words correctly matters. Hoping I haven't committed any faux pas here. :-)

  2. Ouch! I hadn't encountered the Case of the Missing -Ly before, but it definitely deserves a place on the Pet Peeves list!

  3. Actually Euclid had four axioms that seem to correspond to reality and one that he wasn't sure of ... And that fifth axiom, if changed gives you math on a curved surface

  4. But you have to be careful how you change it. One replacement gives you spherical geometry, another gives you hyperbolic geometry.

    And life can get very interesting indeed when you move on to trigonometry. I remember writing a program to calculate when a given point on the earth's surface would be in view of a given satellite. The first version beeped, "Saturday morning! Saturday morning! Saturday...." in an infinite loop. Whoops - used planar trig, which formulas were still in my head, instead of spherical trig.

    Subsequent years of such programming destroyed my ability to do Euclidean trig in my head.


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